84 IV ADDITIONAL PROPERTIES OF THE REAL LINE

bn = cn/2 (when x can be written in two different ways as J} cn/3n, show that the two

n = 0

00

corresponding numbers ]£4,/2tt are equal). Prove that /is a surjective continuous mapping

of K onto the interval [0, 1] of R, and show that K and R are equipotent. Further-
more it is possible to extend /to a continuous mapping of I = [0, 1] onto itself, which is
constant in each of the connected components (3.19.6) of I — K.

3. (a) Let E be a metric space satisfying the following condition : for each finite sequence
•?= fe)iĞcf^n whose terms are equal to 0 or 1, there is a nonempty subset As such that:
(i) E is the union of the two subsets A(0) , A(1), and for each finite sequence s of //
terms, if s', s" are the two sequences of n + I terms whose first n terms are those of s,
AS = AS> u A,-;

(ii) for each infinite sequence (sn)n > i whose terms are equal to 0 or 1 , if sn = (£;)j ^ , <„ ,
the diameter of ASn tends to 0 when n tends to +00, and the intersection of the ASn is
not empty.

Under these conditions, show that there exists a continuous mapping of the triadic
Cantor set K (Problem 2) onto E, and in particular E is compact.

(b) Conversely, let E be an arbitrary compact metric space. Show that there exists a
continuous mapping of K onto E. (Apply the method of (a), and the definition of pre-
compact spaces (Section 3.16); observe that properties (i) and (ii) do not imply that the
two sets AS' and As* need be different from As for all sequences s.)

(c) If in addition E is totally disconnected, and has no isolated points (3.10.10),
then E is homeomorphic to K. (First prove that for every e > 0 there is a
covering of E by a finite number of sets A* which are both open and closed and have a
diameter < s; to that purpose use Problem 9(a) of Section 3.19. Then apply the method
of (a).)

4. (a) Let E (resp. F) be the set of even (resp. odd) natural integers; if, to each subset X
of N, one associates the pair (X n E, X n F), show that one defines a bijection of
$(N) onto $(E) x $(F).

(b) Deduce from (a) and from Problem 2(b) that R" and R are equipotent for all
n > \ (but see Section 5.1, Problem 6).

5. Let I be the interval [0, 1 ] in R. Show that there exists a continuous mapping / of I
onto the "square" I x I (a "Peano curve"). (First show that there is a continuous
mapping of the Cantor set K onto I x I (Problem 3), and then extend the mapping by
linearity to the connected components of the complement of KL in I.) v"'

6. Let g be a mapping of the interval ]0, 1] into the interval [—1,1], and suppose that

lim g(x) = 0. Show that there exist a continuous decreasing mapping g{ and a

jc-*0, x>0

continuous increasing mapping g2 of [0, 1] into [—1,1], such that #i(0) — #2(0) = 0,
and #i(x) < g(x) ^ g2(x) for 0 < x ^ 1 , (For each integer n, consider the g.l.b. xn of the
set of points x such that g(x) ^ I//?.)

3. LOGARITHMS AND EXPONENTIALS

(4.3.1) For any number a> I, there is a unique increasing mapping f of
R* = ]0, + oo[ into R such that f(xy) =/(*) +f(y) andf(a) = 1; moreover,
fis a homeomorphism o/R* onto R. sequence in E), it is necessary and sufficient that for